On the number of vertices of random polyhedra with a given number of facets

The set of points ${\bf x} = (x_1 , \cdots ,x_n )$ satisfying the linear inequalities $\sum\nolimits_{\nu = 1}^n {a_{\mu \nu } } x_\nu \leqq 1\, (\mu = 1, \cdots ,m)$ is a convex polyhedron. If the m points ${\bf a}_\mu = (a_{\mu 1} , \cdots ,a_{\mu n} )$ are chosen independently and uniformly from the unit sphere in n-space, the number $V_{mn} $ of vertices of the polyhedron is a random variable. We give an asymptotic expansion of the expected value $EV_{mn} $ as $m \to \infty $ and an explicit formula for $EV_{mn} $ for any m and n.